{
  "id": "2105.09804",
  "version": "v1",
  "published": "2021-05-20T14:59:02.000Z",
  "updated": "2021-05-20T14:59:02.000Z",
  "title": "Non-existence of measurable solutions of certain functional equations via probabilistic approaches",
  "authors": [
    "Kazuki Okamura"
  ],
  "comment": "7 pages, to appear in Aequationes mathematicae",
  "categories": [
    "math.CA",
    "math.PR"
  ],
  "abstract": "This paper deals with functional equations in the form of $f(x) + g(y) = h(x,y)$ where $h$ is given and $f$ and $g$ are unknown. We will show that if $h$ is a Borel measurable function associated with characterizations of the uniform or Cauchy distributions, then there is no measurable solutions of the equation. Our proof uses a characterization of the Dirac measure and it is also applicable to the arctan equation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-20T14:59:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "39B22",
      "62E10"
    ],
    "keywords": [
      "functional equations",
      "measurable solutions",
      "probabilistic approaches",
      "non-existence",
      "paper deals"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}